Evaluate the following integral:

Solution:

$I=\int \frac{5}{\left(x^{2}+1\right)(x+2)}$

$\frac{5}{\left(x^{2}+1\right)(x+2)}=\frac{A x+B}{x^{2}+1}+\frac{C}{x+2}$

$5=(A x+B)(x+2)+C\left(x^{2}+1\right)$

Equating constants

$5=2 B+C$

Equating coefficients of $x$

$0=2 A+B$

Equating coefficients of $x^{2}$

$0=A+C$

Solving, we get

$A=-1, B=2, C=1$

Thus

$I=\int \frac{-x+2}{x^{2}+1} d x+\int \frac{d x}{x+2}$

$=\int \frac{-\mathrm{xdx}}{\mathrm{x}^{2}+1}+2 \int \frac{\mathrm{dx}}{\mathrm{x}^{2}+1}+\int \frac{\mathrm{dx}}{\mathrm{x}+2}$

$I=-\frac{1}{2} \log \left|x^{2}+1\right|+2 \tan ^{-1} x+\log |x+2|+C$